Escape Rate and Conditional Escape Rate From a Probabilistic Point of View

Abstract

We prove that for a sequence of nested sets \(\{U_n\}\) with \(\Lambda = \cap _n U_n\) a measure zero set, the localized escape rate converges to the extremal index of \(\Lambda \), provided that the dynamical system is \(\phi \)-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns. Examples include a dichotomy for periodic and non-periodic points, Cantor sets on the interval, and submanifolds of Anosov diffeomorphisms on surfaces.